MAT101 SLP3
The most common business application involving exponents is the calculation of compound interest. Let’s introduce some nomenclature.
PV = present value ; the money you pay into the plan today.
t = the duration of an investment program, in years.
FV = future value ; that is, the money you expect to have after n years.
i = interest rate, per year.
I = interest earned
We’ll only discuss the case of a one-time, lump-sum investment. The more common situation, in the real world, is to make more than one payment; usually, payments at regular intervals for the duration of the plan. But those calculations are more complex, so we’ll limit our attention to single-payment plans.
First, we consider simple interest. You make a lump sum payment, which is the Present Value, PV. At the end of n years, you receive your money back (PV) plus interest, calculated as the PV, multiplied by the interest rate, multiplied by the number of years. This is the Future Value (FV) The formula is —
FV = PV(1+it)
Suppose you deposit $1000 at 5% (rate: 0.05) simple interest for ten years. What can you expect to get back, at the end of that period of time?
FV = 1000(1+it)= 1000(1+0.05(10)) = 1000(1.5)= 1500
Questions:
1. You deposit $1000 at 2% for 20 years. What’s the future value?
2. What would be a better deal: depositing money at simple interest of 5% for 20 years, or 10% for 10 years (double the interest, half the time?)
Next, we consider compound interest. This is a more realistic scenario; you make a single deposit, but at the end of every year, the earned interest is added to the principle. Over the years, you earn interest on interest. The formula for compound interest is
FV = PV(1+i)t
Example: What’s the future value of $100 invested at 5% interest, compounded annually, at the end of the first year (n=1)?
FV = 100.00(1+0.05)1=100.00(1.05)=$105.00
That’s simple enough; we could probably have done that in our heads. But how much would $100 be worth, if left on deposit at 5% simple interest, for 20 years?
FV = 100.00(1+0.05)20=100.00(2.65)=$265.00
Where did the 2.65 come from? Almost all calculators will calculate the value of a number raised to a power. If you don’t know how to do it on your calculator, get out the instruction book. Or find a scientific calculator online — there are plenty of them.
3. You deposit $1000 at 2% for 20 years, compounded annually. What’s the future value? Compare with problem 1.
You may have heard your Credit Union advertise, “Interest compounded DAILY on savings accounts!” That means the future value is updated on a daily basis. Instead of n=1 per year, n=365 per year. But you’re certainly not going to get 5% per day. The interest per compounding period (one day) is the annual rate, divided by the number of days in a year. So an annual rate of 5% becomes a daily rate of (5/365)%, or 0.000137.
Let’s see what the difference is, if our $100 investment at 5% per annum is compounded daily for one year.
FV=100.00(1+0.000137)365=100.00(1.000137)365=$105.13
4. You deposit $1000 at 2% for 20 years, compounded daily. What’s the future value? Compare with problem 3. (Remember, in this case your interest is going to be 0.02/365).
5. Suppose you pay $1000 into a savings account that pays 2% per year, compounded annually. How many years will would it take for the money in the account to double, to $2000? (There are formulas for this, but I want you to use “successive approximation.” Guess a number(of years), calculate the FV , then try again. When you find a value of t that gives you between $1800 and $2200, then quit. That’s close enough.)
